To answer your question, yes, I do believe that you could take out the current internal battery from your Red Rescue Team cartridge and replace it with the dead battery of your Emerald game. I specify your Emerald game because if you did decide to do that, you can only replace it with your Emerald game because they are both Gameboy Advance cartridges. Gameboy Advance cartridges require a CR1616 Lithium Cell battery, whereas the Gameboy Color cartridges require either the CR2025 or CR2032 batteries.

Would that be practical? No. You don't know whether or not the battery in your Red Rescue Team is on the verge of drying out anytime soon. I'd suggest just buying new batteries for both your Emerald and Gold cartridges and replacing them yourself. The batteries themselves are not expensive and there a ton of tutorials online that guide you through changing the batteries. You'll also need some simple tools that you probably already have lying around your house.